+171 Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$
+129609 Note that the sum of the first n terms = n * a1 + d * (n)(n-1) / 2 where n > 1
So
S10 = 10a1 + d (10)(9) / 2 = 10a1 + 45d = 0 (1)
S20 = 20a1 + d(20)(19) / 2 = 20a1 + 190d = 1/5 (2)
Multiply (1) by -2 and add to (2) giving
100d = 1/5
d = 1/500
To find a1
10a1 + 45 (1/500) = 0
10a1 = -45/500
a1 = -45 / 5000 = -9 / 1000
So
S70 = 70 (-9/1000) + (1/500) (70)(69) / 2 = 21 / 5 = 4.2
